Question

a table of f(x) is shown below:
x -2 -1 0 1 2
f(x) 2 1 -1 -2 -3
a table for h(x) is shown below:
x -2 -1 0 1 2
h(x) 1 0 -2 -3 -4
based on the table, h(x)=
f(x)+1
f(x+1)
f(x)-1
f(x-1)

Answer 1

To determine the relationship between \( h(x) \) and \( f(x) \), we can evaluate which of the given options matches the values in the table for \( h(x) \).

First, let's look at the values in the tables given:

For \( f(x) \):
- \( f(-2) = 2 \)
- \( f(-1) = 1 \)
- \( f(0) = -1 \)
- \( f(1) = -2 \)
- \( f(2) = -3 \)

For \( h(x) \):
- \( h(-2) = 1 \)
- \( h(-1) = 0 \)
- \( h(0) = -2 \)
- \( h(1) = -3 \)
- \( h(2) = -4 \)

Now let's evaluate the options provided one by one:

1. **\( h(x) = f(x) + 1 \)**

- \( h(-2) = f(-2) + 1 = 2 + 1 = 3 \) (not equal to 1)
- \( h(-1) = f(-1) + 1 = 1 + 1 = 2 \) (not equal to 0)
- \( h(0) = f(0) + 1 = -1 + 1 = 0 \) (not equal to -2)
- \( h(1) = f(1) + 1 = -2 + 1 = -1 \) (not equal to -3)
- \( h(2) = f(2) + 1 = -3 + 1 = -2 \) (not equal to -4)

This option is incorrect.

2. **\( h(x) = f(x+1) \)**

- \( h(-2) = f(-1) = 1 \) (matches)
- \( h(-1) = f(0) = -1 \) (not equal to 0)
- \( h(0) = f(1) = -2 \) (matches)
- \( h(1) = f(2) = -3 \) (matches)
- \( h(2) = f(3) \) (not available)

This option is incorrect.

3. **\( h(x) = f(x) - 1 \)**

- \( h(-2) = f(-2) - 1 = 2 - 1 = 1 \) (matches)
- \( h(-1) = f(-1) - 1 = 1 - 1 = 0 \) (matches)
- \( h(0) = f(0) - 1 = -1 - 1 = -2 \) (matches)
- \( h(1) = f(1) - 1 = -2 - 1 = -3 \) (matches)
- \( h(2) = f(2) - 1 = -3 - 1 = -4 \) (matches)

This option is correct.

4. **\( h(x) = f(x-1) \)**

- \( h(-2) = f(-3) \) (not available)
- \( h(-1) = f(-2) = 2 \) (not equal to 0)
- \( h(0) = f(-1) = 1 \) (not equal to -2)
- \( h(1) = f(0) = -1 \) (not equal to -3)
- \( h(2) = f(1) = -2 \) (not equal to -4)

This option is incorrect.

Thus, the relationship that holds true is:

### \( h(x) = f(x) - 1 \)